In this paper, we investigate the asymptotic regularity of the minimal pullback attractor of a non-autonomous quasi-linear parabolic $p$-Laplacian equation with dynamical boundary condition. First, we establish the higher-order integrability of the difference of solutions near the initial time. Then we show that, under the assumption that the time-depending forcing terms only satisfy some $L^2$ integrability, the $L^2(Ω)× L^2(\partialΩ)$ pullback $\mathscr{D}$-attractor can actually attract the $L^2(Ω)× L^2(\partialΩ)$-bounded set in $L^{2+δ}(Ω)× L^{2+δ}(\partialΩ)$-norm for any $δ∈[0,∞)$.